@Article{Gu2014,
author="Gu, Ran
and Huang, Fei
and Li, Xueliang",
title="Randic incidence energy of graphs",
journal="Transactions on Combinatorics",
year="2014",
volume="3",
number="4",
pages="1-9",
abstract="Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$ and edge set $E(G) = \{e_1, e_2,\ldots, e_m\}$. Similar to the Randi\'c matrix, here we introduce the Randi\'c incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $n\times m$ matrix whose $(i,j)$-entry is $(d_i)^{-\frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randi\'c incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randic incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all $n$-vertex trees. As a result, some results are very different from those for incidence energy.",
issn="2251-8657",
doi="10.22108/toc.2014.5573",
url="http://toc.ui.ac.ir/article_5573.html"
}
@Article{Mathad2014,
author="Mathad, Veena
and Narayankar, Kishori P.",
title="On Lict sigraphs",
journal="Transactions on Combinatorics",
year="2014",
volume="3",
number="4",
pages="11-18",
abstract="A signed graph (marked graph) is an ordered pair $S=(G,\sigma)$‎ ‎$(S=(G,\mu))$‎, ‎where $G=(V,E)$ is a graph called the underlying‎ ‎graph of $S$ and $\sigma:E\rightarrow\{+,-\}$‎ ‎$(\mu:V\rightarrow\{+,-\})$ is a function‎. ‎For a graph $G$‎, ‎$V(G)‎, ‎E(G)$ and $C(G)$ denote its vertex set‎, ‎edge set and cut-vertex‎ ‎set‎, ‎respectively‎. ‎The lict graph $L_{c}(G)$ of a graph $G=(V,E)$‎ ‎is defined as the graph having vertex set $E(G)\cup C(G)$ in which‎ ‎two vertices are adjacent if and only if they correspond to‎ ‎adjacent edges of $G$ or one corresponds to an edge $e_{i}$ of $G$‎ ‎and the other corresponds to a cut-vertex $c_{j}$ of $G$ such that‎ ‎$e_{i}$ is incident with $c_{j}$‎. ‎In this paper‎, ‎we introduce lict‎ ‎sigraphs‎, ‎as a natural extension of the notion of lict graph to‎ ‎the realm of signed graphs‎. ‎We show that every lict sigraph is‎ ‎balanced‎. ‎We characterize signed graphs $S$ and $S^{'}$ for which‎ ‎$S\sim L_{c}(S)$‎, ‎$\eta(S)\sim L_{c}(S)$‎, ‎$L(S)\sim L_{c}(S')$‎, ‎$J(S)\sim L_{c}(S^{'})$ and $T_{1}(S)\sim L_{c}(S^{'})$‎, ‎where‎ ‎$\eta(S)$‎, ‎$L(S)$‎, ‎$J(S)$ and $T_{1}(S)$ are negation‎, ‎line graph‎, ‎jump graph and semitotal line sigraph of $S$‎, ‎respectively‎, ‎and‎ ‎$\sim$ means switching equivalence‎.",
issn="2251-8657",
doi="10.22108/toc.2014.5627",
url="http://toc.ui.ac.ir/article_5627.html"
}
@Article{Chellathurai2014,
author="Chellathurai, S. Robinson
and Vijaya, S. Padma",
title="The geodetic domination number for the product of graphs",
journal="Transactions on Combinatorics",
year="2014",
volume="3",
number="4",
pages="19-30",
abstract="A subset $S$ of vertices in a graph $G$ is called a geodetic set if every vertex not in $S$ lies on a shortest path between two vertices from $S$‎. ‎A subset $D$ of vertices in $G$ is called dominating set if every vertex not in $D$ has at least one neighbor in $D$‎. ‎A geodetic dominating set $S$ is both a geodetic and a dominating set‎. ‎The geodetic (domination‎, ‎geodetic domination) number $g(G) (\gamma(G),\gamma_g(G))$ of $G$ is the minimum cardinality among all geodetic (dominating‎, ‎geodetic dominating) sets in $G$‎. ‎In this paper‎, ‎we show that if a triangle free graph $G$ has minimum degree at least 2 and $g(G) = 2$‎, ‎then $\gamma _g(G) = \gamma(G)$‎. ‎It is shown‎, ‎for every nontrivial connected graph $G$ with $\gamma(G) = 2$ and $diam(G) > 3$‎, ‎that $\gamma_g(G) > g(G)$‎. ‎The lower bound for the geodetic domination number of Cartesian product graphs is proved‎. ‎Geodetic domination number of product of cycles (paths) are determined‎. In this work‎, ‎we also determine some bounds and exact values of the geodetic domination number of strong product of graphs‎.",
issn="2251-8657",
doi="10.22108/toc.2014.5750",
url="http://toc.ui.ac.ir/article_5750.html"
}
@Article{FalahatiNezhad2014,
author="Falahati Nezhad, Farzaneh
and Iranmanesh, Ali
and Tehranian, Abolfazl
and Azari, Mahdieh",
title="Comparing the second multiplicative Zagreb coindex with some graph invariants",
journal="Transactions on Combinatorics",
year="2014",
volume="3",
number="4",
pages="31-41",
abstract="‎‎The second multiplicative Zagreb coindex of a simple graph $G$ is‎ ‎defined as‎: ‎$${\overline{\Pi{}}}_2\left(G\right)=\prod_{uv\not\in{}E(G)}d_G\left(u\right)d_G\left(v\right),$$‎ ‎where $d_G\left(u\right)$ denotes the degree of the vertex $u$ of‎ ‎$G$‎. ‎In this paper‎, ‎we compare $\overline{{\Pi}}_2$-index with‎ ‎some well-known graph invariants such as the Wiener index‎, ‎Schultz‎ ‎index‎, ‎eccentric connectivity index‎, ‎total eccentricity‎, ‎eccentric-distance sum‎, ‎the first Zagreb index and coindex and the‎ ‎first multiplicative Zagreb index and coindex‎.",
issn="2251-8657",
doi="10.22108/toc.2014.5951",
url="http://toc.ui.ac.ir/article_5951.html"
}
@Article{Meemark2014,
author="Meemark, Yotsanan
and Sriwongsa, Songpon",
title="Perfect state transfer in unitary Cayley graphs over local rings",
journal="Transactions on Combinatorics",
year="2014",
volume="3",
number="4",
pages="43-54",
abstract="‎In this work‎, ‎using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra‎, ‎we characterize which local rings allowing PST occurring in its unitary Cayley graph‎. ‎Moreover‎, ‎we have some developments when $R$ is a product of local rings‎.",
issn="2251-8657",
doi="10.22108/toc.2014.5974",
url="http://toc.ui.ac.ir/article_5974.html"
}
@Article{Tavakoli2014,
author="Tavakoli, Mostafa
and Rahbarnia, F.
and Mirzavaziri, M.
and Ashrafi, A. R.",
title="Complete solution to a conjecture of Zhang-Liu-Zhou",
journal="Transactions on Combinatorics",
year="2014",
volume="3",
number="4",
pages="55-58",
abstract="‎‎Let $d_{n,m}=\big[\frac{2n+1-\sqrt{17+8(m-n)}}{2}\big]$ and‎ ‎$E_{n,m}$ be the graph obtained from a path‎ ‎$P_{d_{n,m}+1}=v_0v_1 \cdots v_{d_{n,m}}$ by joining each vertex of‎ ‎$K_{n-d_{n,m}-1}$ to $v_{d_{n,m}}$ and $v_{d_{n,m}-1}$‎, ‎and by‎ ‎joining $m-n+1-{n-d_{n,m}\choose 2}$ vertices of $K_{n-d_{n,m}-1}$‎ ‎to $v_{d_{n,m}-2}$‎. ‎Zhang‎, ‎Liu and Zhou [On the maximal eccentric‎ ‎connectivity indices of graphs‎, ‎Appl‎. ‎Math‎. ‎J‎. ‎Chinese Univ.‎, ‎in‎ ‎press] conjectured that if $d_{n,m}\geqslant 3$‎, ‎then $E_{n,m}$‎ ‎is the graph with maximal eccentric connectivity index among all‎ ‎connected graph with $n$ vertices and $m$ edges‎. ‎In this note‎, ‎we‎ ‎prove this conjecture‎. ‎Moreover‎, ‎we present the graph with‎ ‎maximal eccentric connectivity index among the connected graphs‎ ‎with $n$ vertices‎. ‎Finally‎, ‎the minimum of this graph invariant‎ ‎in the classes of tricyclic and tetracyclic graphs are computed‎.",
issn="2251-8657",
doi="10.22108/toc.2014.5986",
url="http://toc.ui.ac.ir/article_5986.html"
}